Wow, it really is utter rubbish, isn't it?

It's not even a correct statement of the Incompleteness Theorems to begin with (for example, the famous "Gödel sentence" is very obviously not equivalent to the Liar Paradox), so if one simply must "start with it" rather than sensibly hurling it aside and moving on to something less intellectually repellent, that's maybe a good place.

Well, I think you're right that reality doesn't amount to a formal system, but really, in dealing with somebody who thinks this silly article is "gospel," your best bet might be to attack its credibility. And the fact is that it's just plain ill-informed from the get-go, and obviously so: the "Liar statement" is undecidable, and the "Gödel sentence" (for any given formal system) is true. Great heavens, that's why the proof works in the first place.

I agree that you should just attack his credibility. Focus on his characterizations of Godel's theorems and point out that they don't seem to be equivalent to Godel's theorem. For instance, ask him to show that the statement in terms of "circles" is equivalent to Godel's statement, or any statement shown by any trained logician.

What is a "closed system"? What are some examples of closed and open systems?

He emphasizes repeatedly the idea that you cannot "prove" anything. He says, "All reasoning ultimately traces back to faith in something that you cannot prove." How does he reconcile that with the fact that Godel, like any logician, takes it to be possible to prove things in incomplete (even in inconsistent) systems?

I rarely pass up an opportunity to recommend Torkel Franzén's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.